Stability of Flocking Motion Herbert G. Tanner, Ali Jadbabaie and George J. Pappas University of Pennsylvania Technical Report No: MS-CIS-03-03 Abstract This paper investigates the aggregated stability properties of of a system of multiple mobile agents described by simple dynamical systems. The agents are steered through local coordinating control laws that arise as a combina- tion of attractive/repulsive and alignment forces. These forces ensure colli- sion avoidance and cohesion of the group and result to all agents attaining a common heading angle, exhibiting flocking motion. Two cases are consid- ered: in the first, position information from all group members is available to each agent; in the second, each agent has access to position information of only the agents laying inside its neighborhood. It is then shown that regard- less of any arbitrary changes in the neighbor set, the flocking motion remains stable as long as the graph that describes the neighboring relations among the agents in the group is always connected. 1 Introduction Over the past decade a considerable amount of attention has been focused on the problem of coordinated motion of multiple autonomous agents. From biological sciences to systems and controls, and from statistical physics to computer graphics, researchers have been trying to develop an understanding of how a group of moving objects such as flocks of birds, schools of fish, crowds of people [35, 20], or man-made mobile autonomous agents, can move in a formation only using local interactions and without a global supervisor. Such problems have been studied in ecology and theoretical biology, in the context of animal aggregation and social cohesion in animal groups [1, 23, 37, 13, 9]. A computer model mimicking animal aggregations was gener- ated in [27]. Following the work in [27] several other computer models have appeared in the literature (cf. [14] and the references therein), and has led to creation of a new area in computer graphics known as artificial life [27, 32]. At the same time, several researchers in the area of statistical physics and complexity theory have addressed flocking and schooling behavior in the context of non-equilibrium phenomena in many-degree-of-freedom dynamical systems and self organization in systems of self-propelled particles [36, 34, 33, 21, 18, 30, 6, 16]. Similar problems have become a major thrust in systems and control theory, in the context of cooperative control, distributed control of multiple vehicles and formation control; see for example [17, 24, 25, 7, 19, 10, 31, 15, 22]. The main goal of the above papers is to develop a distributed control strategy for each individual vehicle or agent, such that the global objec- tive, such as a tight formation with fixed pair-wise inter vehicle distances is achieved. In 1986 Craig Reynolds [27] made a computer model of coordinated an- imal motion such as bird flocks and fish schools. It was based on three dimensional computational geometry of the sort normally used in computer animation or computer aided design. He called the generic simulated flocking creatures “boids”. The basic flocking model consists of three simple steering behaviors which describe how an individual boid maneuvers based on the positions and velocities its nearby flockmates: • Separation: steer to avoid crowding local flockmates. • Alignment: steer towards the average heading of local flockmates. 1 • Cohesion: steer to move toward the average position of local flock- mates. Each boid has direct access to the whole scene’s geometric description, but flocking requires that it reacts only to flockmates within a certain small neighborhood around itself. The neighborhood is characterized by a distance (measured from the center of the boid) and an angle, measured from the boid’s direction of flight. Flockmates outside this local neighborhood are ignored. The neighborhood could be considered a model of limited perception (as by fish in murky water) but it is probably more correct to think of it as defining the region in which flockmates influence a boids steering. The super position of these three rules results in all boids moving in a formation, while avoiding collision. Several variation on this model were also proposed. An example of such generalizations is a leader follower strategy, in which one agent acted as a group leader and the other boids would just follow the same rules as before. Simulations indicate that in this case, all agents tend to follow the leader. In 1995, a similar model was proposed by Vicsek et al. [36]. Vicsek’s model, though developed independently, turns out to be a special case of the latter model, in which all agents move with the same speed (no dynamics), and only follow an alignment rule. In this scenario, each agents heading is updated as the average of the headings of agent itself with its nearest neighbors plus some additive noise. Numerical simulations in [36] indicate the spontaneous development of coherent collective motion, resulting in the headings of all agents to converge to a common value. This was quite a surprising result in the physics community and was followed by a series of papers [5, 34, 33, 28, 21]. A proof of convergence for Vicsek’s model (in the noise-free case) was given in [15]. The goal of this paper is to develop a mathematical model for the flocking phenomenon in [27], and provide a system theoretic justification by combin- ing results from classical control theory, mechanics, algebraic graph theory, nonsmooth analysis and Lyapunov stability for nonsmooth systems . We will show that the cohesion and separation rules can be decoupled from alignment. It will be shown that under assumptions on connectivity of the graph rep- resenting the nearest neighbor relation, all agents headings converge to the same value, and all velocities will eventually become the same. Moreover, all pairwise distances converge, resulting in a flocking behavior. In [15], similar nearest neighbor alignment control laws have been pro- 2 posed and the stability of the resulting flocking motion was analyzed. The models used in [15] to describe the agent dynamics were kinematic, whereas collision avoidance was not an issue. Further, in [15], it was also shown that stability can be recovered even when connectivity in the network of agent is temporarily lost. While the proof techniques are totally different from those in [15], the end result is similar, suggesting that addition of cohesion and sep- aration forces in addition to alignment as well as addition of dynamics, does not affect the stability of the flocking motion. In the end, all that is needed for flocking to happen is connectivity in the network of agents, although the analysis in this paper requires connectivity to be maintained at all times. This paper is organized as follows: in Section 2 we describe the system, the problem that we are addressing and we highlight the approach that we are going to follow. Section 3 introduces some basic facts in algebraic graph theory that are used in the stability analysis. The first case, in which each agent uses state information from all other agents in the group, is treated in Section 4. Then, Section 5 presents some important results from Lya- punov stability theory for nonsmooth systems. The case where only local information is used in the control law of each agent is addressed in Section 6. Section 7 follows with numerical simulations, verifying our stability results and Section 8 summarizes and points out our main contributions. 2 Problem Formulation and Overview Consider N agents, moving on the plane (generalization to three dimensions is straightforward) with the following dynamics: r˙i = vi (1a) v˙i = ai i =1,...,N , (1b) T T where ri =(xi,yi) is the position of agent i, vi =(˙xi, y˙i) is its velocity and T ai =(ux,uy) its control (acceleration) inputs. Define the orientation angle (heading) of agent i, θi,as: θi = arctan2(y ˙i, x˙ i) . (2) The relative positions between the agents are denoted rij = ri − rj. Each agent is steered via its acceleration input with the objective being to achieve group coordination through local, decentralized control action. The 3 acceleration input consists of two components (Figure 1): ai = ari + aθi . (3) The first component, ari , is the negated vector field of an artificial potential function, Vi that encodes the relative position of agent i with respect to its neighbors and is designed to have a minimum at a desired configuration or relative distances between them and to increase when agent i comes very close to its neighbors or moves too far away from them. Term ari , therefore, is thought of ensuring collision avoidance and cohesion in the group. The second component, aθi is assumed that it regulates the heading of agent i to the weighted average of that of its neighbors by rotating its velocity vector. Acting in a direction normal to the agent velocity, these control forces do not contribute to the kinetic energy of the system and can be regarded as gyroscopic [2]. vi aθi θi yˆ ri −∇ ri Vi zˆ xˆ Figure 1: Control forces acting on agent i. The objective is to determine the control input components so that the group exhibits a stable, collision free flocking motion. This is being under- stood technically as a convergence property on the heading differences and the relative distances of the agents in the group. In the following Sections it will be shown that for a very generic class of artificial potential functions, stable flocking motion can be established using local control schemes. Stability can still be established even when the set of neighboring agents providing feedback information to a particular 4 agent controller, changes arbitrarily during motion.